Skip to main content
SpringerLink
Log in

Optimal H1 spatial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

A time-fractional Allen-Cahn problem is considered, where the spatial domain Ω is a bounded subset of \(\mathbb {R}^{d}\) for some d ∈{1,2,3}. New bounds on certain derivatives of the solution are derived. These are used in the analysis of a numerical method (L1 discretization of the temporal fractional derivative on a graded mesh, with a standard finite element discretization of the spatial diffusion term, and Newton linearization of the nonlinear driving term), showing that the computed solution achieves the optimal rate of convergence in the Sobolev H1(Ω) norm. (Previous papers considered only convergence in L2(Ω).) Numerical results confirm our theoretical findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, R.A., Fournier, J.: Cone conditions and properties of Sobolev spaces. J. Math. Anal. Appl. 61(3), 713–734 (1977). https://doi.org/10.1016/0022-247X(77)90173-1. https://doi-org.ucc.idm.oclc.org/10.1016/0022-247X(77)90173-1

    Article  MathSciNet  Google Scholar 

  2. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015). https://doi.org/10.1016/j.jcp.2014.09.031

    Article  MathSciNet  MATH  Google Scholar 

  3. Bramble, J.H., Pasciak, J.E., Steinbach, O.: On the stability of the L2 projection in H1(Ω). Math. Comput. 71(237), 147–156 (2002). https://doi.org/10.1090/S0025-5718-01-01314-X

    Article  Google Scholar 

  4. Diethelm, K.: The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, vol. 2004. Springer, Berlin (2010). An application-oriented exposition using differential operators of Caputo type

    Google Scholar 

  5. Du, Q., Yang, J., Zhou, Z.: Time-fractional Allen-Cahn equations: analysis and numerical methods. arXiv:1906.06584 (2019)

  6. Elliott, C.M., Larsson, S.: Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation. Math. Comp. 58(198), 603–630, S33–S36 (1992). https://doi.org/10.2307/2153205

    Article  MathSciNet  MATH  Google Scholar 

  7. Ganesan, S., Tobiska, L.: Finite Elements. Theory and Algorithms. Cambridge University Press, Delhi (2017)

    Book  Google Scholar 

  8. Huang, C., Stynes, M.: A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition. Appl. Numer. Math. 135, 15–29 (2019). https://doi.org/10.1016/j.apnum.2018.08.006

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, C., Stynes, M.: Superconvergence of a finite element method for the multi-term time-fractional diffusion problem. J. Sci. Comput. 82(1), Paper No. 10, 17 (2020). https://doi.org/10.1007/s10915-019-01115-w

    Article  MathSciNet  MATH  Google Scholar 

  10. Jin, B., Li, B., Zhou, Z.: Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56(1), 1–23 (2018). https://doi.org/10.1137/16M1089320

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, D., Wu, C., Zhang, Z.: Linearized Galerkin FEMs for nonlinear time fractional parabolic problems with non-smooth solutions in time direction. J. Sci. Comput. 80(1), 403–419 (2019). https://doi.org/10.1007/s10915-019-00943-0

    Article  MathSciNet  MATH  Google Scholar 

  12. McLean, W.: Regularity of solutions to a time-fractional diffusion equation. ANZIAM J. 52 (2), 123–138 (2010). https://doi.org/10.1017/S1446181111000617

    Article  MathSciNet  MATH  Google Scholar 

  13. Ren, J., Huang, C., An, N.: Direct discontinuous Galerkin method for solving nonlinear time fractional diffusion equation with weak singularity solution. Appl. Math. Lett. 102, 106,111 (2020). https://doi.org/10.1016/j.aml.2019.106111

    Article  MathSciNet  Google Scholar 

  14. Ren, J., Liao, H.L., Zhang, J., Zhang, Z.: Sharp H1-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems. Preprint, arXiv:1811.08059 (2018)

  15. Stynes, M.: Singularities. In: Handbook of Fractional Calculus with Applications, vol. 3, pp 287–305. De Gruyter, Berlin (2019)

  16. Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057–1079 (2017). https://doi.org/10.1137/16M1082329

    Article  MathSciNet  MATH  Google Scholar 

  17. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. 2Nd Revised and Expanded Ed., 2Nd Revised and Expanded Ed. Edn. Springer, Berlin (2006)

    Google Scholar 

Download references

Acknowledgments

We thank an unknown reviewer for pointing out an error in the analysis of our original paper. This work was completed while Chaobao Huang was visiting Beijing CSRC.

Funding

The research of Chaobao Huang is supported in part by the National Natural Science Foundation of PR China (Grant Nos. 11801332 and 11971276). The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under grant NSAF-U1930402.

Author information

Authors and Affiliations

  1. School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, 250014, People’s Republic of China

    Chaobao Huang

  2. Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing, 100193, People’s Republic of China

    Martin Stynes

Authors
  1. Chaobao Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Stynes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Stynes.

Additional information

Communicated by: Lourenco Beirao da Veiga

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, C., Stynes, M. Optimal H1 spatial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation. Adv Comput Math 46, 63 (2020). https://doi.org/10.1007/s10444-020-09805-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-020-09805-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation